1. use the factor theorem
looking at 24, I tried ±1, ±2, ...
and found that x=-1 and x=-2 worked
so I used synthetic division consecutively and found
x^4+10x^3+35x^2+50x+24
= (x+1)(x+2)(x^2 + 7x + 12)
and the quadratic factors again, so the final factors are
(x+1)(x+2)(x+3)(x+4)